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In 1938, Tarski proved that a formula is not intuitionistically valid if, andonly if, it has a counter-model in the Heyting algebra of open sets of sometopological space. In fact, Tarski showed that any Euclidean space R^n with n>= 1 suffices, as does e.g. the Cantor space. In particular, intuitionisticlogic cannot detect topological dimension in the frame of all open sets of aEuclidean space. By contrast, we consider the lattice of open subpolyhedra of agiven compact polyhedron P \subseteq R^n, prove that it is a locally finiteHeyting subalgebra of the (non-locally-finite) algebra of all open sets of R^n,and show that intuitionistic logic is able to capture the topological dimensionof P through the bounded-depth axiom schemata. Further, we show thatintuitionistic logic is precisely the logic of formul{\ae} valid in all Heytingalgebras arising from polyhedra in this manner. Thus, our main theoremreconciles through polyhedral geometry two classical results: topologicalcompleteness in the style of Tarski, and Jaskowski's theorem thatintuitionistic logic enjoys the finite model property. Several questions ofinterest remain open. E.g., what is the intermediate logic of all closedtriangulable manifolds?

Tarski's Theorem on Intuitionistic logic, for polyhedra